Physics for Geologists, Second edition

Heat and heat flow 73
The quantity of heat, Q [ML2TP2], that flows through an area A and
thickness 1 in time t is given by
where k is the thermal conductivity and A0 is the difference of temperature
across thickness I. We are usually more interested in heat per unit of time
and per unit area, so
The equation is properly written with a minus sign before the k to indicate
the direction of flow towards smaller energies. The larger the thermal
conductivity, the smaller the geothermal gradient. The dimensions of k are
(ML~T-~)(T-')(L-~)(L) = MLT-3, per K or "C; in units, Wm-I K-'.
For all rocks, the thermal conductivity is very small but different lithologies
have different conductivities. The heat flow unit is mW mP2; but the older
literature has units of pcal/cm2 sec, which is equal to 41.87 mW m-2.
Within the Earth there are surfaces of equal temperature, called isothermal
surfaces. Heat flow is normal to these surfaces, just as water flow is normal
to surfaces of equal energy (equipotential surfaces).
The geothermal gradient (which is the thermal gradient vertically into the
Earth) in a homogeneous rock is
Ae - Q = 4 [~-'e].
1 (KtA) k
The product of the geothermal gradient and the thermal conductivity is the
heat flow, q (in w m-2), more properly called the density of heat flow rate.
But a sequence is made up of rocks of different conductivities, and so if
possible we should take them all into account, with their thicknesses:
Typically, sedimentary rocks have thermal conductivities between 1 and
3 w m-I K-' (see Clark 1966: 459-82) while for frozen soil with 20 per
cent moisture content it is about 0.2 W m-' K-'.
Permafrost and icecaps
There is a special case of interest to do with ice. In the polar regions there
are both icecaps and, in places, thick sequences of rocks in which the pore
water remains frozen all year below a depth of a metre or so. Such frozen
rocks are called permafrost, and in some areas of North America the per-
mafrost reaches thicknesses of 600 to 700 m, and in Siberia even double that
Copyright 2002 by Richard E. Chapman